Vol. 7, No. 2, 2013

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Base change behavior of the relative canonical sheaf related to higher dimensional moduli

Zsolt Patakfalvi

Vol. 7 (2013), No. 2, 353–378
Abstract

We show that the compatibility of the relative canonical sheaf with base change fails generally in families of normal varieties. Furthermore, it always fails if the general fiber of a family of pure dimension n is Cohen–Macaulay and the special fiber contains a strictly Sn1 point. In particular, in moduli spaces with functorial relative canonical sheaves Cohen–Macaulay schemes can not degenerate to Sn1 schemes. Another, less immediate consequence is that the canonical sheaf of an Sn1, G2 scheme of pure dimension n is not S3.

Keywords
canonical sheaf, relative canonical sheaf, dualizing complex, relative dualizing complex, base change, depth, moduli of stable varieties
Mathematical Subject Classification 2010
Primary: 14J10
Secondary: 14D06, 14F10, 14E30
Milestones
Received: 9 June 2011
Revised: 26 January 2012
Accepted: 9 March 2012
Published: 25 April 2013
Authors
Zsolt Patakfalvi
Department of Mathematics
Princeton University
Fine Hall, Washington Road
Princeton, NJ 08540
United States
http://www.math.princeton.edu/~pzs